280 Mr Dixon, On Lie's Solution of a Partial 



Then we know that the integration of any partial differential 

 equation 



f(x lt Xo, ...x n , z,p 1} p. 2 ,...p n ) = (1) 



depends on the solution of the linear equation 



(f><f>) = (2) 



for $ in terras of x x , . . . x n , z, p x . . . p n . The general solution of this 

 linear equation is derivable when that of the equation f= is 

 known. 



§ 3. With regard to the expression (/, <f>) two things are 

 noticeable. In the first place it is indifferent whether we do, or 

 do not, suppose that wherever p n , say, occurs in </> it is to be 

 regarded as a known function of the other 2n variables, denned by 

 the relation f= 0. This is true for each term in the expression 

 and therefore for the whole. 



Secondly, the following relation holds 

 (u, (v, w)) + (v, (w, u)) + (w, (u, v)) 



= _ (vw) __ (WjW) __ (M) „)_ (3 ). 



Hence if u, v, w are all solutions of the equation (/, </>) = it 

 follows that the ratios 



(v, w) : (w, u) : (u, v) 



are also solutions, unless they are constants. 



§ 4. If now we suppose p n to be defined as a function of 



Xx ... X n , Z, p 1 ... p n —i 



by the equation f= 0, and therefore not to occur in <f> except 

 apparently, the equation (/, <f>) = 0, which is linear and homo- 

 geneous in the 2n derivatives of <f>, is satisfied by 2?i — 1 independent 

 functions, say u 1} u 2 ••• u-2n-i> an( i the most general form of <£ is a 

 function of these. 



We take any such form, say w 1; and seek next a common 

 solution of the equations 



(/,<£) = 0, K,0) = O. 



This must be a function of u l} u 2 ... w 2n _ 1 , such that 



S (u 1 ,u r )~-=0. 

 . = i ou r 



The coefficients in this equation, or at least their ratios, are 

 functions of u 1} u 2 ... u^-! since they satisfy the equation 



